Acoustic theory - AbsoluteAstronomy.com

Acoustic theory is the field relating to mathematical description of sound

Sound

Sound is a mechanical wave that is an oscillation of pressure transmitted through a solid, liquid, or gas, composed of frequencies within the range of hearing and of a level sufficiently strong to be heard, or the sensation stimulated in organs of hearing by such vibrations.-Propagation of...

waves

WAVES

The WAVES were a World War II-era division of the U.S. Navy that consisted entirely of women. The name of this group is an acronym for "Women Accepted for Volunteer Emergency Service" ; the word "emergency" implied that the acceptance of women was due to the unusual circumstances of the war and...

. It is derived from fluid dynamics

Fluid dynamics

In physics, fluid dynamics is a sub-discipline of fluid mechanics that deals with fluid flow—the natural science of fluids in motion. It has several subdisciplines itself, including aerodynamics and hydrodynamics...

. See acoustics

Acoustics

Acoustics is the interdisciplinary science that deals with the study of all mechanical waves in gases, liquids, and solids including vibration, sound, ultrasound and infrasound. A scientist who works in the field of acoustics is an acoustician while someone working in the field of acoustics...

for the engineering

Engineering

Engineering is the discipline, art, skill and profession of acquiring and applying scientific, mathematical, economic, social, and practical knowledge, in order to design and build structures, machines, devices, systems, materials and processes that safely realize improvements to the lives of...

approach.

The propagation of sound waves in a fluid (such as water) can be modeled by an equation of motion (conservation of momentum

Momentum

In classical mechanics, linear momentum or translational momentum is the product of the mass and velocity of an object...

) and an equation of continuity (conservation of mass

Mass

Mass can be defined as a quantitive measure of the resistance an object has to change in its velocity.In physics, mass commonly refers to any of the following three properties of matter, which have been shown experimentally to be equivalent:...

). With some simplifications, in particular constant density, they can be given as follows:

where

is the acoustic pressure and

is the acoustic fluid velocity vector,

is the vector of spatial coordinates

is the time,

is the static mass density of the medium and

is the bulk modulus

Bulk modulus

The bulk modulus of a substance measures the substance's resistance to uniform compression. It is defined as the pressure increase needed to decrease the volume by a factor of 1/e...

of the medium. The bulk modulus can be expressed in terms of the density and the speed of sound in the medium (

) as

The acoustic wave equation

Acoustic wave equation

In physics, the acoustic wave equation governs the propagation of acoustic waves through a material medium. The form of the equation is a second order partial differential equation. The equation describes the evolution of acoustic pressure p or particle velocity u as a function of position r and...

is a combination of these two sets of balance equations and can be expressed as

The acoustic wave equation

Acoustic wave equation

(and the mass and momentum balance equations) are often expressed in terms of a scalar potential

Scalar potential

A scalar potential is a fundamental concept in vector analysis and physics . The scalar potential is an example of a scalar field...

where

. In that case the acoustic wave equation

Acoustic wave equation

is written as

and the momentum balance and mass balance are expressed as

Derivation of the governing equations

The derivations of the above equations for waves in an acoustic medium are given below.

Conservation of momentum

The equations for the conservation of linear momentum for a fluid medium are

where

is the body force per unit mass,

is the pressure, and

is the deviatoric stress. If

is the Cauchy stress, then

where

is the rank-2 identity tensor.

We make several assumptions to derive the momentum balance equation for an acoustic medium. These assumptions and the resulting forms of the momentum equations are outlined below.

Assumption 1: Newtonian fluid

In acoustics, the fluid medium is assumed to be Newtonian

Newtonian fluid

A Newtonian fluid is a fluid whose stress versus strain rate curve is linear and passes through the origin. The constant of proportionality is known as the viscosity.-Definition:...

. For a Newtonian fluid, the deviatoric stress tensor is related to the velocity by

where

is the shear viscosity

Viscosity

Viscosity is a measure of the resistance of a fluid which is being deformed by either shear or tensile stress. In everyday terms , viscosity is "thickness" or "internal friction". Thus, water is "thin", having a lower viscosity, while honey is "thick", having a higher viscosity...

and

is the bulk viscosity.

Therefore, the divergence of

is given by

Using the identity

, we have

The equations for the conservation of momentum may then be written as

Assumption 2: Irrotational flow

For most acoustics problems we assume that the flow is irrotational, that is, the vorticity is zero. In that case

and the momentum equation reduces to

Assumption 3: No body forces

Another frequently made assumption is that effect of body forces on the fluid medium is negligible. The momentum equation then further simplifies to

Assumption 4: No viscous forces

Additionally, if we assume that there are no viscous forces in the medium (the bulk and shear viscosities are zero), the momentum equation takes the form

Assumption 5: Small disturbances

An important simplifying assumption for acoustic waves is that the amplitude of the disturbance of the field quantities is small. This assumption leads to the linear or small signal acoustic wave equation. Then we can express the variables as the sum of the (time averaged) mean field (

) that varies in space and a small fluctuating field (

) that varies in space and time. That is

and

Then the momentum equation can be expressed as

Since the fluctuations are assumed to be small, products of the fluctuation terms can be neglected (to first order) and we have

Assumption 6: Homogeneous medium

Next we assume that the medium is homogeneous; in the sense that the time averaged variables

and

have zero gradients, i.e.,

The momentum equation then becomes

Assumption 7: Medium at rest

At this stage we assume that the medium is at rest which implies that the mean velocity is zero, i.e.

. Then the balance of momentum reduces to

Dropping the tildes and using

, we get the commonly used form of the acoustic momentum equation

Conservation of mass

The equation for the conservation of mass in a fluid volume (without any mass sources or sinks) is given by

where

is the mass density of the fluid and

is the fluid velocity.

The equation for the conservation of mass for an acoustic medium can also be derived in a manner similar to that used for the conservation of momentum.

Assumption 1: Small disturbances

From the assumption of small disturbances we have

and

Then the mass balance equation can be written as

If we neglect higher than first order terms in the fluctuations, the mass balance equation becomes

Assumption 2: Homogeneous medium

Next we assume that the medium is homogeneous, i.e.,

Then the mass balance equation takes the form

Assumption 3: Medium at rest

At this stage we assume that the medium is at rest, i.e.,

. Then the mass balance equation can be expressed as

Assumption 4: Ideal gas, adiabatic, reversible

In order to close the system of equations we need an equation of state

Equation of state

In physics and thermodynamics, an equation of state is a relation between state variables. More specifically, an equation of state is a thermodynamic equation describing the state of matter under a given set of physical conditions...

for the pressure. To do that we assume that the medium is an ideal gas

Ideal gas

An ideal gas is a theoretical gas composed of a set of randomly-moving, non-interacting point particles. The ideal gas concept is useful because it obeys the ideal gas law, a simplified equation of state, and is amenable to analysis under statistical mechanics.At normal conditions such as...

and all acoustic waves compress the medium in an adiabatic and reversible manner. The equation of state can then be expressed in the form of the differential equation:

where

is the specific heat at constant pressure,

is the specific heat at constant volume, and

is the wave speed. The value of

is 1.4 if the acoustic medium is air.

For small disturbances

where

is the speed of sound

Speed of sound

The speed of sound is the distance travelled during a unit of time by a sound wave propagating through an elastic medium. In dry air at , the speed of sound is . This is , or about one kilometer in three seconds or approximately one mile in five seconds....

in the medium.

Therefore,

The balance of mass can then be written as

Dropping the tildes and defining

gives us the commonly used expression for the balance of mass in an acoustic medium:

Governing equations in cylindrical coordinates

If we use a cylindrical coordinate system

Cylindrical coordinate system

A cylindrical coordinate system is a three-dimensional coordinate systemthat specifies point positions by the distance from a chosen reference axis, the direction from the axis relative to a chosen reference direction, and the distance from a chosen reference plane perpendicular to the axis...

with basis vectors

, then the gradient

Gradient

In vector calculus, the gradient of a scalar field is a vector field that points in the direction of the greatest rate of increase of the scalar field, and whose magnitude is the greatest rate of change....

and the divergence

Divergence

In vector calculus, divergence is a vector operator that measures the magnitude of a vector field's source or sink at a given point, in terms of a signed scalar. More technically, the divergence represents the volume density of the outward flux of a vector field from an infinitesimal volume around...

are given by

where the velocity

Velocity

In physics, velocity is speed in a given direction. Speed describes only how fast an object is moving, whereas velocity gives both the speed and direction of the object's motion. To have a constant velocity, an object must have a constant speed and motion in a constant direction. Constant ...

has been expressed as

.

The equations for the conservation of momentum may then be written as

In terms of components, these three equations for the conservation of momentum in cylindrical coordinates are

The equation for the conservation of mass

Conservation of mass

The law of conservation of mass, also known as the principle of mass/matter conservation, states that the mass of an isolated system will remain constant over time...

can similarly be written in cylindrical coordinates as

Time harmonic acoustic equations in cylindrical coordinates

The acoustic equations for the conservation of momentum and the conservation of mass

Conservation of mass

The law of conservation of mass, also known as the principle of mass/matter conservation, states that the mass of an isolated system will remain constant over time...

are often expressed in time harmonic

Harmonic

A harmonic of a wave is a component frequency of the signal that is an integer multiple of the fundamental frequency, i.e. if the fundamental frequency is f, the harmonics have frequencies 2f, 3f, 4f, . . . etc. The harmonics have the property that they are all periodic at the fundamental...

form (at fixed frequency

Frequency

Frequency is the number of occurrences of a repeating event per unit time. It is also referred to as temporal frequency.The period is the duration of one cycle in a repeating event, so the period is the reciprocal of the frequency...

). In that case, the pressures and the velocity are assumed to be time harmonic functions of the form

where

is the frequency. Substitution of these expressions into the governing equations in cylindrical coordinates gives us the fixed frequency form of the conservation of momentum

and the fixed frequency form of the conservation of mass

Conservation of mass

The law of conservation of mass, also known as the principle of mass/matter conservation, states that the mass of an isolated system will remain constant over time...

Special case: No z-dependence

In the special case where the field quantities are independent of the z-coordinate we can eliminate

to get

Assuming that the solution of this equation can be written as

we can write the partial differential equation as

The left hand side is not a function of

while the right hand side is not a function of

. Hence,

where

is a constant. Using the substitution

we have

The equation on the left is the Bessel equation which has the general solution

where

is the cylindrical Bessel function

Bessel function

In mathematics, Bessel functions, first defined by the mathematician Daniel Bernoulli and generalized by Friedrich Bessel, are canonical solutions y of Bessel's differential equation:...

of the first kind and